Converting Pi to binary: Don't do it
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This is an "illusion" of sorts that works by ignoring contextual information. Sure, all that stuff is in pi, but it takes intelligence for someone to extract the right number of bits starting at the right position and decide to put that in a certain context where it makes sense (SSN, executable machine code, JPEG, etc.).
For example, the UTF-8 representation of this comment is somewhere in pi, but you can't say that the full information of this comment is in pi. Part of the information lies outside of pi, in the person who finds that part of pi and chooses to decode it as UTF-8.
It's because you can find absolutely anything there that you can never be sure that you've found what you're looking for.
The first problem with that is finding the desired content somewhere in the number. You can't just Google it. In fact, I know of no way to search for it except for brute force (i.e. continue to calculate pi until such time as you find it). Because the odds are so low for any reasonably sized file, this is going to take a long time. Even if you measure using a scale where one unit is the current age of the universe, you would find the scale to be too small.
Even if someone solves that problem, the other problem is file size. You have to store offset and length. The length for most files would be reasonable, the problem is the offset. In all probability, that number is larger than your original file. Much larger. There are exceptions, of course, but not enough of them to help.
That said, I wonder how long it will be before some mathematician starts writing theorems about the "set of illegal numbers" like the 09F9 number and the rest.
I think the real problem, as you said, is having a copy of the full sequence around for encoding and decoding, and finding any given string of bits within it.
Furthermore, I would guess the offset has even less entropy on average than the bitsequence you're hoping to compress, so if you had an algorithm that would shorten the expression of the offset you might very well be able to apply it to the original bitsequence with the same or better results.
I'd think an information theory expert could probably tackle these sorts of questions very rigorously, but I am not one so this is mostly conjecture.
Namely that there are 2^n possible signals with n bits, so if a compression algorithm compresses some signal that is larger than n bits down to n bits or less, then there aren't enough possibilities left for all possible signals of at most n to also be encoded with n bits or less.
However, you'll still find that there are still limits to exactly how much you can compress things. And you'll find that, sometimes when you think you've found something that looks like it should be able to compress things down to nothing, that you've just been hiding the data in your decompression program.
In a sense, when you consider special-purpose compression and decompression functions, it's not unlike how "RETR some_huge_file.rar" sent to an FTP program will "decompress" that tiny string into some multi-GB file. But that only works because the program already has a copy of the data.
Figure as an upper bound that there are 10^90 particles in the universe. Now imagine that you could turn the entire universe into such an efficient computer that every particle stored a bit of information, and you used it to calculate pi.
The probability of finding an arbitrary binary string of length N in that pi is (1/2^N)*10^90. Set to .5, solve for N, and you get . . .
388 bits. You have a 50/50 probability of finding an arbitrary 388 bit string somewhere in there. For reference, that's about 50 characters in ASCII. Maybe enough for a threat against the president. CD cracking software seems more dubious.
When dealing with brute force probability, the numbers get big very, very fast. Don't assume that just because they seem big to you means they're big enough to solve the problem you're thinking about. Do the math.
(don't trust pi!)
I wonder if we can even extract a full sentence that makes sense out of the current digits that we have.
1)We are living in an illusion and are simulated (not like matrix, as in a real program) on the computer of a geeky alien in his basement (or equivalent in his world).
2)A method is discovered to capture bit strings which are part of real world and thus creates a computer which knows all the secrets.
3)That we are not the makers of our own destinies. Pi holds our past present and future.
Man this is exciting exploration into science fiction.
Instead the Library is a place where you have books that contain the every string of symbols.
http://jubal.westnet.com/hyperdiscordia/library_of_babel.htm...
Still not convinced that mining Pi for usable data would be a bad idea? How about this: You will be able to find those recipes not only once, you will find them several times. As ASCII encoded bytes, as PDF document, as DOC document, as JPEG, as iPhone and Android app (different versions for every version of iOS and Android that there will ever be). And you don’t have to stop there, you can just continue on. Just think of the millions of versions of the recipes with a few typos.
1)A genius mathematicians.
2)A theorem to extract contextual information from Pi bits. It would give not only the info, but also work out the context.
Yes, the plans involved Pi.